Transceivers with decision feedback equalizers - Signal Processing and Optimization for Transceiver Systems - page 595

implemented sequentially within a block in a block-based communication sys-
tem such as, for example, the zero-padded or the cyclic-prefix system. The
general form of the system is shown in Fig. 19.3. Comparing this with the linear
transceiver of Fig. 16.1, we see that the only new element here is the matrix B ,
called the feedback matrix , which provides a feedback path from the detected
symbols s est ( n )tothe input of the detector. The matrix G will henceforth be
called the feedforward matrix. The matrix B is chosen to be strictly upper tri-
angular , that is, upper triangular with zeros on the main diagonal, so that the
signal v is related to s est as follows:

⎡ ⎤ ⎡ ⎤ ⎡ ⎤

0 b 01 b 02 ... b 0 ,M− 1 s 0 ,est

s 1 ,est
.
s M− 2 ,est
s M − 1 ,est

v v
.
v M− 2
v M − 1

00 b 12 ... b 1 ,M− 1

⎣ ⎦ ⎣ ⎦ ⎣ ⎦

. . . .

. . .

= . (19 . 5)

00 0 ... b M− 2 ,M− 1

00 0 ... 0

v

s est

For example if M =3wehave
⎡ ⎤ ⎡ ⎤ ⎡ ⎤

v v v 2 0 b 01 b 02

00 b 12
00 0

s 0 ,est
s 1 ,est
s 2 ,est

⎣ ⎦ = ⎣ ⎦ ⎣ ⎦ ,

(19 . 6)

and the details of the feedback part are shown in Fig. 19.4. In this case v 2 =0.
The generation of v 1 and v 0 are detailed in the figure. Though it is not indicated
in the figure, the operation of the structure should be interpreted sequentially ,
from bottom to top, as follows:

1. First, the bottom-most detector input s 2 ( n ) is used to obtain the estimate

s 2 ,est ( n )of s 2 ( n ) directly.

2. This estimate s 2 ,est ( n ) is then used to correct the input s 1 ( n ) to the next

detector, and the estimate s 1 ,est ( n )of s 1 ( n )isproduced.

3. The estimates s 2 ,est ( n )and s 1 ,est ( n ) are then used to correct the input

s 0 ( n ) to the next detector (topmost in this case). This detector then

produces the estimate s 0 ,est ( n )of s 0 ( n ).

Summarizing, we start from the bottom-most signal and proceed to the top.
When a symbol estimate s k,est ( n ) is produced, it is used (along with the esti-
mates s k +1 ,est ( n ) ,...,s M− 1 ,est ( n )) to correct the signal

s k− 1 ( n ) and produce the
estimate s k− 1 ,est ( n ). Thus, even though the feedback structure in Figs. 19.3 and
19.4 appear to have delay-free loops, the actual operation ( M steps described
above) follows sequentially within one block-processing interval. It is clear that
the strict upper triangular nature of the matrix B is crucial for the successful
operation of this receiver structure.